Mastering College Algebra: Solving Quadratic Equations with Ease

When it comes to mastering College Algebra, solving quadratic equations can feel like a daunting task. But fear not! Let's break down the equation (3y^2 - 5xy + 2x^2 = 0), ensuring you grasp the concepts needed for your CLEP exam prep.

First off, what does this equation mean? Essentially, we're on a quest to find the values of (y) that keep this equation balanced. Think of it as a see-saw. Both sides need to hold steady, and we’re simply trying to discover the sweet spot.

To start solving, we can factor the equation. Factorization is kind of like pulling apart a puzzle to see how all the pieces come together. By rearranging our equation into ((3y - 2x)(y - x) = 0), we’re laying out the pieces neatly. But what does this mean for us? It means we can break down our problem into more manageable sections.

Now, let’s tackle these two factors:

1. First Factor: (3y - 2x = 0)
2. Second Factor: (y - x = 0)

By solving the first factor, we find (y = \frac{2}{3}x). Plugging this back into our equation keeps the balance, confirming it's a legitimate solution.

On to the second factor, (y = x). While this seems appealing, we hit a snag! In the context of our equation, we can't have (y) and (x) being the same. So, guess what? This leaves us with just one clear winner from our options: (y = \frac{2}{3}x)! Isn’t that satisfying?

Now, what about those other answer choices? Let’s quickly debunk them to keep our understanding solid:

• Option A: If (y = 0), substituting back into the equation makes it equal to zero, but doesn’t truly solve for our (y).

• Option B: The suggestion (y = -\frac{2}{3}x) doesn’t line up with our factored equation either. A quick check shows it leads us astray.

• Option D: Saying there's no solution simply isn’t right. We proved that there’s a solution right before our eyes!

This isn’t just about finding the right answer; it’s about understanding the steps that lead us there. Getting comfortable with these concepts, like factoring and setting up equations, will lay down the groundwork for many algebra-related quests you might face—be it in an exam hall or in everyday applications.

Remember, practice makes perfect! The more you play around with these concepts, the more they’ll stick. As you prep for your College Algebra CLEP exam, keep tackling problems like this, and approach each challenge with a curious mindset. You’re building critical thinking skills that will serve you well beyond just a single test.

So, whether it’s solving quadratic equations or diving into other algebra realms, maintain that inquisitive spirit! Now, let’s get cracking on some more practice problems to reinforce what we've learned today!